Statistical fluctuations of matrix elements in regular and chaotic systems

Abstract

A combination of semiclassical arguments and random-matrix theory is used to analyze transition strengths in quantum systems whose associated classical systems are chaotic. The mean behavior is found semiclassically while the local fluctuations are characterized by a Porter-Thomas distribution. The methods are tested numerically for a system with two degrees of freedom, the coupled-rotators model. The deviations of the strength distribution from a Porter-Thomas one when the system is nonchaotic are also investigated. It is found that the distribution gets gradually wider as the classical system becomes more regular. The transition of a classical system from regular to chaotic behavior is fairly well understood. The manifestations of this classical behavior in the associate quantum system has been the subject of many investigations in recent years with particular attention given to the chaotic regime. The few quantitative results are based either on semiclassical arguments or on random-matrix theory. For nonintegrable systems there is no working scheme for semiclassical quantization. The Einstein-Brillouin-Keller quantization breaks down in those regions of phase space where invariant tori do not exist. On the other hand, the extent to which random-matrix theory can be applied in chaotic regions is still not fully understood. In this paper we shall argue that a reasonable scheme for the analysis of a quantum system whose associate classical motion is chaotic involves a combination of semiclassical and random-matrix methods. In this scheme the mean behavior of the respective quantity is given semiclassically, while the local Auctuations are reproduced via random-Inatrix theory. Such a procedure had been invoked for the energy spectrum. The average level density is found semiclassically or by an empirical procedure and the spectrum is then renormalized by dividing out the mean spacing. The unfolded spectrum has then the statistical properties characterizing an ensemble of Gaussian orthogonal random matrices (GOE), as was found numerically for several chaotic systems. In particular, a Wigner distribution is obtained for the levelspacing distribution. A similar procedure emerges for the mean behavior and Auctuation properties ' of transition-matrix elements. In particular transition strength distributions seem to be characterized locally by Porter-Thomas distributions. The purpose of this paper is to establish this procedure for the matrix elements and to study the deviations from a Porter-Thomas distribution as the classical system becomes more regular. The procedure consists of two steps. First, the average behavior of the transition strengths is found semiclassically or empirically. Second, the actual transition strengths are renormalized by dividwhere ~i ) and ~f ) are eigenstates of the Hamiltonian with eigenvalues E, and F&, respectively. One can easily show that S is the Fourier transform of the temporal autocorrelation function C(t) of T g(E E') = e' *C(t)dt, 27TA (2)